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Sample Paths and Limit Theorems
Published in Randal Douc, Eric Moulines, David S. Stoffer, Nonlinear Time Series, 2014
Randal Douc, Eric Moulines, David S. Stoffer
We conclude this chapter by introducing some concentration type inequalities. Concentration of measure is a fairly general phenomenon which, roughly speaking, asserts that a Borel function fX0,X1,…,Xn-1 with suitably small local oscillations almost always takes values that are close to the average (or median) value ExfX0,X1,…,Xn-1. Under various assumptions on the function f, this phenomenon has been quite extensively studied when X0,X1,…,Xn are i.i.d. The situation is naturally far more complex for nonproduct measures, where one can trivially construct examples where the concentration property fails. For functions of dependent random variables Xt,t∈N, the crux of the problem is often to quantify and bound the dependence among the random variables in terms of various types of mixing coefficients. For simplicity, we focus on uniformly ergodic Markov chains (Definition 6.7); some recent results will allow us to extend such bounds to more general V-geometric Markov chains, but these results are technically much more involved. We prove a form of the McDiarmid bounded difference inequality [see (7.62)] for uniformly ergodic Markov chains, starting from any initial distribution. These results are illustrated using examples from nonparametric statistics.
A new water quality index elaborated under the brazilian legislation perspective
Published in International Journal of River Basin Management, 2022
Alana Deduck Cicilinski, Jorim Sousa Virgens Filho
WQINSF is an index composed by nine parameters (Dissolved Oxygen (DO), Fecal Coliforms (FC), pH, Biological Oxygen Demand (BOD), Nitrates, Total phosphates, Temperature, Turbidity and Total Solids (TS)), where each one has an assigned weight (w), as shown in Table 3. In addition to its weight, each parameter has a quality value (q), obtained from a graph according to its concentration or measure. The calculation of the WQINSF is performed by the production of the weighted outcome of the nine parameters, according to Equation (1).where: WQINSF= Water Quality Index; qi =quality of the ith parameter, obtained from the respective quality chart (further details on the ‘q’ values obtained by quality graphs for each parameter can be found at http://home.eng.iastate.edu/~dslutz/dmrwqn/water_quality_index_calc.htm); wi = weight corresponding to the ith parameter set due to its importance for the overall conformation of quality.
Temporal and spatial statistical analysis of ambient air quality of Assam (India)
Published in Journal of the Air & Waste Management Association, 2020
Gouri Sankar Bhunia, Ding Ding
RMSE is the recurrently used to measure the degree of the variances between values prophesied by a model or an estimator and the values observed. It is principally representing the sample standard deviation of the modifications between projected and observed. RMSE offers significant information in predicting the extent of pollutant concentration, a measure adjacent to zero embodies virtuous prediction. The absolute mean percentage error signified by MAPE is intended by dividing sum of percentage error by number of observations, a value equal to or close to zero is measured as perfect. The coefficient of efficiency indicates the normalized fit of the model, the value ranges between -∞ to 1 (Nash and Sutcliffe 1970). The value 1 signposts a seamless fit. Ross (1996) proposed a simple multiplicative factor called accuracy factor (ACFT) representing the spread of the modeled data. RMSE, MAPE, and r2 were primarily used as performance measure; RMSE < 8 MAPE < 25% and r2 > 0.5 were considered a good measure (Shareef, Husain, and Alharbi 2016).
Research of O2O website based consumer purchase decision-making model
Published in Journal of Industrial and Production Engineering, 2019
The meaning of the F value is used to test the results of the sample to represent the true degree of the population. The F test is also called the homogeneity test of variance. It is used to identify whether the two population variances are equal. The P value refers to the probability that the F test or other test quantity is greater than the value obtained and it is generally less than the given value indicates that the test is significant. The M value represents the average concentration trend measure of a set of data. The T test, also known as the Student’s t test, is mainly used for a normal distribution with small sample content (eg, n < 30) and an overall standard deviation σ unknown. The T test uses the t-distribution theory to infer the probability of .a difference occurring [46]. Same as all the parameters below.